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PROBABILITY THEORY
Probability is a measure of uncertainty. Example:
- Drawing a number at random between 0 and 1 has probability 0. So for each number it has 0 probability and combinational form will give 0 probability as a whole.
- P(Number drawn is ) between 0 and 1.
So, P(Number drawn is 1/2).
Rules of Probability.
In an experiment let be the set of all outcomes. is called the Sample space.
A subset A of is called an event. Event A occurs if any occurs. To define probability measures we need to specify probabilities of a collection of events,. @ is just a specified collection of subsets of the set. Only sets in will be assigned probabilities.
Q. What properties must have?
Answer : be closed under finite unions and finite intersections of set in as well as under complementation
An induction argument shows that if are sets in then so are.
i.e., By, ……(i)
Also if …….(ii)
So from (i) & (ii)
A non−empty collection satisfying (i) & (ii) is called a field of subsets of or an algebra of subsets of.
However an algebra is not sufficient to define a probability measure.
−field or −algebra: A non−empty collection of subsets of a set is called a −filed of subsets of provided that the following twp properties hold. (i) If A. (ii) If is a sequence of sets in , then
both belong to.
Probability Measure:
A probability measure P on a −algebra or −field of subset of is a function such that,
i) ii) P iii) If is a sequence of mutually disjoint (exclusive) sets in , then
Let then
Power sets of
Properties of Probability Measures:
If , then
(i) For
(ii) (iii) (By D’morgan’s Law)
Since
Using (1) & (2), we have
And consequently
Ex:
Solution :
Homework:
Can be proved by induction on A.
Exercise : 200 people attended a dinner party.
of them ate potato salad (among other things)
of them ate chicken casserole.
of them had potato salad but not chicken casserole.
What proportions ate either potato salad or chicken casserole (or both)?
Now,
So, using (i) & (ii)
Finite Probability : In a finite probability model, the sample space has a finite number of outcomes. Say
Define simple events:
Ex: Toss 2 coins together
Sample Random Sampling
Consider a population with N,.
We want to choose a random sample of n units from this population to study its characteristics. This is a random experiments where the outcomes are samples of n units from the above population.
Q. A bowl has 100 marbles of which 50 are red, 30 blue and 20 green. Consider drawing 5 marbles at random from this bowl. How are marbles drawn? How does the composition of the bowl change after each draw?
Simple Random Sampling with replacement (SRSWR)
(i) Each marble in the bowl at the time of drawing has the same chance of being drawn at each draw. After each draw, the color of the marble drawn is noted and the marble drawn is returned to the bowl. In this sampling composition does not change.
Simple Random Sampling without Replacement (SRSWOR)
Each marble in the bowl at the time of drawing has the chance of being drawn at each draw.
After each draw, the color of the marble drawn is noted. But the marble drawn is not returned to the bowl. * i.e., contents of the bowl changes.
Ordered Sample:
If a set S has m distinct points and another set T has n distinct points, then the numbers of pairs that can be formed is mn.
If we take then no. of k−tuples is.
Where (S).
SRSWR:
Population has N distinct units. A sample of size n is drawn using SRSWR.
All the outcomes are equally likely. Each simple event has probability is.
If an event E consist of k outcomes, then
Exercise: Roll a pair of balanced die once. What is the probability that the sum of the dots which show up is 7?
Solution:
Possible favorable outcome
SRSWOR is a discrete uniform probability model also but the sample is an ordered samples (n−tuples) where any unit can appear at most once.
Permutation: A permutation of n distinct objects is an arrangement of these in a particular order, say, two objects.
A, B can be arranged as.
So, total i.e., 21.
Similarly, {A, B, C} can be arranged in 6 ways i.e, 31.
Number of distinguishable permutation of n object is
Consider a set S of N distinct elements. Select one object from it. Without replacing it to the set, draw one from the remaining (N−1) objects. Continue until we have a sample of n objects..
Outcome is an n−tuple
Total no. of such outcomes or samples is
Al outcomes are equally likely simple event has probability
Exercise: Consider n distinct boxes and n distinct balls. There are n! ways of throwing these balls into boxes such that each box gets one ball. In such an experiment probability that ball numbered getting placed into box numbered j is.
Arguments for this:
Q. What is the probability a team of 5 consist of 3 men & 2 women is
Conditional Probability
Toss a fair 10 times. Probability of any event is ratio of the number of outcomes favouring the event to 2^10 (total no. of outcomes).
Q. What is the probability that the last toss came up head?
Q. What is the probability that the last toss came up heads given that there was at least one head in the experiment?
Now we know that the sample space include (TT….TT), but all others are exactly likely.
P(last toss came up heads given that the experiment resulted in at least one head)
Definition:
Let A & B be two events such that P(A) > 0. Then the conditional probability of B given A is defined as
Ex: Toss a fair coin 10 times. What is the probability of getting exactly 5 heads?
Solution: Total outcome
Possible favorable outcome
Ex: A bowl has m marbles of which m 1 , are red and the rest blue, n marbles are drawn at random without replacing. What is the probability that there will be k red marbles in the sample?
Solution:
Ex: For any day in September suppose the following probability model holds.
Bright Sunshine
Partly Sunny
Totally cloud.
Rain
,
(a) Given that it is cloudy, what is the probability that will rain?
Each outcome in belongs one and only one of the partitioning events.
For any event B,
Consider E such that P(E) > 0 and take as the partition of. Then
Bayes Theorem: If P(A) > 0 and P (B)> 0. Then
Proof:
Generalizes to
Ex: In a particular community of the people smoke. For a smoker there is a chance of getting cancer, whereas for a non−smoker it is 20. What is the chance that a cancer patient chosen at random is a smoker?
Solution:
Ex: Toss a fair coin 3 times. Find
(a) P(2nd^ toss comes up head) (b) P(2nd^ toss comes up head | 1st^ cost coming up head)
Solution:
(a)
(b)
A gives no information about the occurrence of B.
Definition: If then we say that A & B are independent events. (probabilistic/statistical).
Definition: Two events A & B are independent if
Ex: Consider SRSWR & SRSWOR of sample size from a set.
Sample Events
{c, c} {(a, a)} {(a, b)} {(a, c)} {(b, a)} {(b, b)} {(b, c)} {(c, a)} {(c, b)}
Probability (WR)
Probability (WOR)
WR: {both draws produce ‘a’}
A, B & C are pair wise independent.
n events are said to be mutually independent if for all subcollections of size.
- Toss a coin 3 times.
- Choose a sample of size 2 using SRSWR for a population of size 100.
- SRSWOR – no independence for draws P
- Roll a die. If you hit a six, roll it again. If no six stop.
Ex: (1): Toss a fair coin 3 times. It is reasonable to assume that outcome of the first toss does not influence that of 2nd^ toss, or 3rd, outcome of 2nd^ does not influence that of 3rd, so mutual independence of components of the experiment may be assured.
for each toss
Ex (2): Draw a sample of size 2 from population of 100 using SRSWR.
In Steps :
Ex(3): Same with SRSWOR
Elements of Combinatorial Analysis
RULE – I : If there are two groups G 1 & G 2 ; consisting of n elements and
consisting of m elements then the no. of pairs formed by taking one element.
If there are k groups , such that
Then the number ordered k−tuples formed by taking one element from each group is
Example : ‘Placing balls into the cells’ amounts to choose one cell for each ball. Let there are r balls and n cells. For the 1st^ ball, we can choose any one of the n cells. Similarly, for each of the balls, we have n choices, assuming the capacity of each cell is infinite or we can place more than one ball in each cell. Hence the r balls can be placed in the n cells in ways.
Applications:
1. A die is rolled r times. Find the probability that – i) No ace turns up. [ace −1] ii) No ace turns up.
Solution:
i) The experiment of throwing a die r times has possible outcomes.
Favorable sample is
Hence, the probability is
Example: 2) If n balls are randomly placed into n cells, what is the probability that each cell will be occupied.
Solution:
SOLVED EXAMPLES:
1. Find the probability that among five randomly selected digits, all digits are different.
Ans:
2. In a city seven accidents occur each week in a particular week there occurs one accidents per day. Is it surprising?
Ans :
3. An elevator (lift) stands with 7 passengers and stops at 10th^ floor. What is the probability that no two passengers leave at the same floor?
Solution:
4. What is the probability that r individuals have different birthdays? Also show that the probability is approximately equal to. How many people are required to make the prob. of distinct birthdays less than ½?
Solution:
More than 23 people are required.
5. Six dice are thrown. What’s the prob. that every possible number will appear.
Hints:
6. There are four children in a family. Find the prob. that (a) At least two of them have the same birthday? (b) Only the oldest and the youngest have the same birthday?
